Properties

Label 6240.2.a.bn.1.2
Level $6240$
Weight $2$
Character 6240.1
Self dual yes
Analytic conductor $49.827$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.8266508613\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6240.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} -0.828427 q^{17} -2.82843 q^{19} -2.82843 q^{21} +2.82843 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.828427 q^{29} -1.65685 q^{31} +4.00000 q^{33} +2.82843 q^{35} -3.65685 q^{37} +1.00000 q^{39} +11.6569 q^{41} +1.65685 q^{43} +1.00000 q^{45} -9.65685 q^{47} +1.00000 q^{49} +0.828427 q^{51} -9.31371 q^{53} -4.00000 q^{55} +2.82843 q^{57} -4.00000 q^{59} -2.00000 q^{61} +2.82843 q^{63} -1.00000 q^{65} -2.82843 q^{69} -8.00000 q^{71} +3.17157 q^{73} -1.00000 q^{75} -11.3137 q^{77} -5.65685 q^{79} +1.00000 q^{81} -15.3137 q^{83} -0.828427 q^{85} -0.828427 q^{87} +3.65685 q^{89} -2.82843 q^{91} +1.65685 q^{93} -2.82843 q^{95} -1.51472 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} - 8 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 8 q^{31} + 8 q^{33} + 4 q^{37} + 2 q^{39} + 12 q^{41} - 8 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} + 4 q^{53} - 8 q^{55} - 8 q^{59} - 4 q^{61} - 2 q^{65} - 16 q^{71} + 12 q^{73} - 2 q^{75} + 2 q^{81} - 8 q^{83} + 4 q^{85} + 4 q^{87} - 4 q^{89} - 8 q^{93} - 20 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −0.828427 −0.200923 −0.100462 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) −2.82843 −0.617213
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.828427 0.153835 0.0769175 0.997037i \(-0.475492\pi\)
0.0769175 + 0.997037i \(0.475492\pi\)
\(30\) 0 0
\(31\) −1.65685 −0.297580 −0.148790 0.988869i \(-0.547538\pi\)
−0.148790 + 0.988869i \(0.547538\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 2.82843 0.478091
\(36\) 0 0
\(37\) −3.65685 −0.601183 −0.300592 0.953753i \(-0.597184\pi\)
−0.300592 + 0.953753i \(0.597184\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 11.6569 1.82049 0.910247 0.414065i \(-0.135891\pi\)
0.910247 + 0.414065i \(0.135891\pi\)
\(42\) 0 0
\(43\) 1.65685 0.252668 0.126334 0.991988i \(-0.459679\pi\)
0.126334 + 0.991988i \(0.459679\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −9.65685 −1.40860 −0.704298 0.709904i \(-0.748738\pi\)
−0.704298 + 0.709904i \(0.748738\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.828427 0.116003
\(52\) 0 0
\(53\) −9.31371 −1.27934 −0.639668 0.768651i \(-0.720928\pi\)
−0.639668 + 0.768651i \(0.720928\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 2.82843 0.374634
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 2.82843 0.356348
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −2.82843 −0.340503
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 3.17157 0.371205 0.185602 0.982625i \(-0.440576\pi\)
0.185602 + 0.982625i \(0.440576\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −11.3137 −1.28932
\(78\) 0 0
\(79\) −5.65685 −0.636446 −0.318223 0.948016i \(-0.603086\pi\)
−0.318223 + 0.948016i \(0.603086\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.3137 −1.68090 −0.840449 0.541891i \(-0.817709\pi\)
−0.840449 + 0.541891i \(0.817709\pi\)
\(84\) 0 0
\(85\) −0.828427 −0.0898555
\(86\) 0 0
\(87\) −0.828427 −0.0888167
\(88\) 0 0
\(89\) 3.65685 0.387626 0.193813 0.981039i \(-0.437915\pi\)
0.193813 + 0.981039i \(0.437915\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) 0 0
\(93\) 1.65685 0.171808
\(94\) 0 0
\(95\) −2.82843 −0.290191
\(96\) 0 0
\(97\) −1.51472 −0.153796 −0.0768982 0.997039i \(-0.524502\pi\)
−0.0768982 + 0.997039i \(0.524502\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 6.48528 0.645310 0.322655 0.946517i \(-0.395425\pi\)
0.322655 + 0.946517i \(0.395425\pi\)
\(102\) 0 0
\(103\) −2.34315 −0.230877 −0.115439 0.993315i \(-0.536827\pi\)
−0.115439 + 0.993315i \(0.536827\pi\)
\(104\) 0 0
\(105\) −2.82843 −0.276026
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 2.48528 0.238047 0.119023 0.992891i \(-0.462024\pi\)
0.119023 + 0.992891i \(0.462024\pi\)
\(110\) 0 0
\(111\) 3.65685 0.347093
\(112\) 0 0
\(113\) −14.4853 −1.36266 −0.681330 0.731976i \(-0.738598\pi\)
−0.681330 + 0.731976i \(0.738598\pi\)
\(114\) 0 0
\(115\) 2.82843 0.263752
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −2.34315 −0.214796
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −11.6569 −1.05106
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.34315 0.207921 0.103960 0.994581i \(-0.466849\pi\)
0.103960 + 0.994581i \(0.466849\pi\)
\(128\) 0 0
\(129\) −1.65685 −0.145878
\(130\) 0 0
\(131\) −10.8284 −0.946084 −0.473042 0.881040i \(-0.656844\pi\)
−0.473042 + 0.881040i \(0.656844\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 13.3137 1.13747 0.568733 0.822522i \(-0.307434\pi\)
0.568733 + 0.822522i \(0.307434\pi\)
\(138\) 0 0
\(139\) −2.34315 −0.198743 −0.0993715 0.995050i \(-0.531683\pi\)
−0.0993715 + 0.995050i \(0.531683\pi\)
\(140\) 0 0
\(141\) 9.65685 0.813254
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 0.828427 0.0687971
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −15.6569 −1.28266 −0.641330 0.767265i \(-0.721617\pi\)
−0.641330 + 0.767265i \(0.721617\pi\)
\(150\) 0 0
\(151\) 20.9706 1.70656 0.853280 0.521453i \(-0.174610\pi\)
0.853280 + 0.521453i \(0.174610\pi\)
\(152\) 0 0
\(153\) −0.828427 −0.0669744
\(154\) 0 0
\(155\) −1.65685 −0.133082
\(156\) 0 0
\(157\) 21.3137 1.70102 0.850510 0.525960i \(-0.176294\pi\)
0.850510 + 0.525960i \(0.176294\pi\)
\(158\) 0 0
\(159\) 9.31371 0.738625
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 16.9706 1.32924 0.664619 0.747183i \(-0.268594\pi\)
0.664619 + 0.747183i \(0.268594\pi\)
\(164\) 0 0
\(165\) 4.00000 0.311400
\(166\) 0 0
\(167\) −15.3137 −1.18501 −0.592505 0.805567i \(-0.701861\pi\)
−0.592505 + 0.805567i \(0.701861\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.82843 −0.216295
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 2.82843 0.213809
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) −8.48528 −0.634220 −0.317110 0.948389i \(-0.602712\pi\)
−0.317110 + 0.948389i \(0.602712\pi\)
\(180\) 0 0
\(181\) 19.6569 1.46108 0.730541 0.682869i \(-0.239268\pi\)
0.730541 + 0.682869i \(0.239268\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −3.65685 −0.268857
\(186\) 0 0
\(187\) 3.31371 0.242322
\(188\) 0 0
\(189\) −2.82843 −0.205738
\(190\) 0 0
\(191\) −19.3137 −1.39749 −0.698745 0.715370i \(-0.746258\pi\)
−0.698745 + 0.715370i \(0.746258\pi\)
\(192\) 0 0
\(193\) −21.7990 −1.56913 −0.784563 0.620049i \(-0.787113\pi\)
−0.784563 + 0.620049i \(0.787113\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) 9.31371 0.663574 0.331787 0.943354i \(-0.392348\pi\)
0.331787 + 0.943354i \(0.392348\pi\)
\(198\) 0 0
\(199\) −16.9706 −1.20301 −0.601506 0.798869i \(-0.705432\pi\)
−0.601506 + 0.798869i \(0.705432\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.34315 0.164457
\(204\) 0 0
\(205\) 11.6569 0.814150
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) 11.3137 0.782586
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) 1.65685 0.112997
\(216\) 0 0
\(217\) −4.68629 −0.318126
\(218\) 0 0
\(219\) −3.17157 −0.214315
\(220\) 0 0
\(221\) 0.828427 0.0557260
\(222\) 0 0
\(223\) −2.82843 −0.189405 −0.0947027 0.995506i \(-0.530190\pi\)
−0.0947027 + 0.995506i \(0.530190\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −6.34315 −0.421009 −0.210505 0.977593i \(-0.567511\pi\)
−0.210505 + 0.977593i \(0.567511\pi\)
\(228\) 0 0
\(229\) −8.82843 −0.583399 −0.291699 0.956510i \(-0.594221\pi\)
−0.291699 + 0.956510i \(0.594221\pi\)
\(230\) 0 0
\(231\) 11.3137 0.744387
\(232\) 0 0
\(233\) 26.4853 1.73511 0.867554 0.497343i \(-0.165691\pi\)
0.867554 + 0.497343i \(0.165691\pi\)
\(234\) 0 0
\(235\) −9.65685 −0.629944
\(236\) 0 0
\(237\) 5.65685 0.367452
\(238\) 0 0
\(239\) 16.9706 1.09773 0.548867 0.835910i \(-0.315059\pi\)
0.548867 + 0.835910i \(0.315059\pi\)
\(240\) 0 0
\(241\) 21.3137 1.37294 0.686468 0.727160i \(-0.259160\pi\)
0.686468 + 0.727160i \(0.259160\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 2.82843 0.179969
\(248\) 0 0
\(249\) 15.3137 0.970467
\(250\) 0 0
\(251\) −5.17157 −0.326427 −0.163213 0.986591i \(-0.552186\pi\)
−0.163213 + 0.986591i \(0.552186\pi\)
\(252\) 0 0
\(253\) −11.3137 −0.711287
\(254\) 0 0
\(255\) 0.828427 0.0518781
\(256\) 0 0
\(257\) −3.17157 −0.197837 −0.0989186 0.995096i \(-0.531538\pi\)
−0.0989186 + 0.995096i \(0.531538\pi\)
\(258\) 0 0
\(259\) −10.3431 −0.642692
\(260\) 0 0
\(261\) 0.828427 0.0512784
\(262\) 0 0
\(263\) −14.1421 −0.872041 −0.436021 0.899937i \(-0.643613\pi\)
−0.436021 + 0.899937i \(0.643613\pi\)
\(264\) 0 0
\(265\) −9.31371 −0.572137
\(266\) 0 0
\(267\) −3.65685 −0.223796
\(268\) 0 0
\(269\) −1.51472 −0.0923540 −0.0461770 0.998933i \(-0.514704\pi\)
−0.0461770 + 0.998933i \(0.514704\pi\)
\(270\) 0 0
\(271\) −9.65685 −0.586612 −0.293306 0.956019i \(-0.594755\pi\)
−0.293306 + 0.956019i \(0.594755\pi\)
\(272\) 0 0
\(273\) 2.82843 0.171184
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −12.6274 −0.758708 −0.379354 0.925252i \(-0.623854\pi\)
−0.379354 + 0.925252i \(0.623854\pi\)
\(278\) 0 0
\(279\) −1.65685 −0.0991933
\(280\) 0 0
\(281\) −26.9706 −1.60893 −0.804464 0.594001i \(-0.797548\pi\)
−0.804464 + 0.594001i \(0.797548\pi\)
\(282\) 0 0
\(283\) −25.6569 −1.52514 −0.762571 0.646905i \(-0.776063\pi\)
−0.762571 + 0.646905i \(0.776063\pi\)
\(284\) 0 0
\(285\) 2.82843 0.167542
\(286\) 0 0
\(287\) 32.9706 1.94619
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) 0 0
\(291\) 1.51472 0.0887944
\(292\) 0 0
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) −2.82843 −0.163572
\(300\) 0 0
\(301\) 4.68629 0.270113
\(302\) 0 0
\(303\) −6.48528 −0.372570
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 2.34315 0.133297
\(310\) 0 0
\(311\) −30.6274 −1.73672 −0.868361 0.495933i \(-0.834826\pi\)
−0.868361 + 0.495933i \(0.834826\pi\)
\(312\) 0 0
\(313\) 10.9706 0.620093 0.310046 0.950721i \(-0.399655\pi\)
0.310046 + 0.950721i \(0.399655\pi\)
\(314\) 0 0
\(315\) 2.82843 0.159364
\(316\) 0 0
\(317\) 9.31371 0.523110 0.261555 0.965189i \(-0.415765\pi\)
0.261555 + 0.965189i \(0.415765\pi\)
\(318\) 0 0
\(319\) −3.31371 −0.185532
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 2.34315 0.130376
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −2.48528 −0.137436
\(328\) 0 0
\(329\) −27.3137 −1.50585
\(330\) 0 0
\(331\) −7.51472 −0.413046 −0.206523 0.978442i \(-0.566215\pi\)
−0.206523 + 0.978442i \(0.566215\pi\)
\(332\) 0 0
\(333\) −3.65685 −0.200394
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 23.6569 1.28867 0.644335 0.764743i \(-0.277134\pi\)
0.644335 + 0.764743i \(0.277134\pi\)
\(338\) 0 0
\(339\) 14.4853 0.786732
\(340\) 0 0
\(341\) 6.62742 0.358895
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) −2.82843 −0.152277
\(346\) 0 0
\(347\) 1.65685 0.0889446 0.0444723 0.999011i \(-0.485839\pi\)
0.0444723 + 0.999011i \(0.485839\pi\)
\(348\) 0 0
\(349\) 15.1716 0.812116 0.406058 0.913847i \(-0.366903\pi\)
0.406058 + 0.913847i \(0.366903\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −2.68629 −0.142977 −0.0714884 0.997441i \(-0.522775\pi\)
−0.0714884 + 0.997441i \(0.522775\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 2.34315 0.124012
\(358\) 0 0
\(359\) 11.3137 0.597115 0.298557 0.954392i \(-0.403495\pi\)
0.298557 + 0.954392i \(0.403495\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 3.17157 0.166008
\(366\) 0 0
\(367\) −2.34315 −0.122311 −0.0611556 0.998128i \(-0.519479\pi\)
−0.0611556 + 0.998128i \(0.519479\pi\)
\(368\) 0 0
\(369\) 11.6569 0.606832
\(370\) 0 0
\(371\) −26.3431 −1.36767
\(372\) 0 0
\(373\) 13.3137 0.689358 0.344679 0.938721i \(-0.387988\pi\)
0.344679 + 0.938721i \(0.387988\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −0.828427 −0.0426662
\(378\) 0 0
\(379\) −19.7990 −1.01701 −0.508503 0.861060i \(-0.669801\pi\)
−0.508503 + 0.861060i \(0.669801\pi\)
\(380\) 0 0
\(381\) −2.34315 −0.120043
\(382\) 0 0
\(383\) 34.6274 1.76938 0.884689 0.466181i \(-0.154371\pi\)
0.884689 + 0.466181i \(0.154371\pi\)
\(384\) 0 0
\(385\) −11.3137 −0.576600
\(386\) 0 0
\(387\) 1.65685 0.0842226
\(388\) 0 0
\(389\) −32.1421 −1.62967 −0.814835 0.579692i \(-0.803173\pi\)
−0.814835 + 0.579692i \(0.803173\pi\)
\(390\) 0 0
\(391\) −2.34315 −0.118498
\(392\) 0 0
\(393\) 10.8284 0.546222
\(394\) 0 0
\(395\) −5.65685 −0.284627
\(396\) 0 0
\(397\) 4.34315 0.217976 0.108988 0.994043i \(-0.465239\pi\)
0.108988 + 0.994043i \(0.465239\pi\)
\(398\) 0 0
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) −30.2843 −1.51232 −0.756162 0.654384i \(-0.772928\pi\)
−0.756162 + 0.654384i \(0.772928\pi\)
\(402\) 0 0
\(403\) 1.65685 0.0825338
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 14.6274 0.725054
\(408\) 0 0
\(409\) −11.6569 −0.576394 −0.288197 0.957571i \(-0.593056\pi\)
−0.288197 + 0.957571i \(0.593056\pi\)
\(410\) 0 0
\(411\) −13.3137 −0.656717
\(412\) 0 0
\(413\) −11.3137 −0.556711
\(414\) 0 0
\(415\) −15.3137 −0.751720
\(416\) 0 0
\(417\) 2.34315 0.114744
\(418\) 0 0
\(419\) −7.51472 −0.367118 −0.183559 0.983009i \(-0.558762\pi\)
−0.183559 + 0.983009i \(0.558762\pi\)
\(420\) 0 0
\(421\) 4.82843 0.235323 0.117662 0.993054i \(-0.462460\pi\)
0.117662 + 0.993054i \(0.462460\pi\)
\(422\) 0 0
\(423\) −9.65685 −0.469532
\(424\) 0 0
\(425\) −0.828427 −0.0401846
\(426\) 0 0
\(427\) −5.65685 −0.273754
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 8.97056 0.432097 0.216048 0.976383i \(-0.430683\pi\)
0.216048 + 0.976383i \(0.430683\pi\)
\(432\) 0 0
\(433\) −19.6569 −0.944648 −0.472324 0.881425i \(-0.656585\pi\)
−0.472324 + 0.881425i \(0.656585\pi\)
\(434\) 0 0
\(435\) −0.828427 −0.0397200
\(436\) 0 0
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 12.9706 0.616250 0.308125 0.951346i \(-0.400298\pi\)
0.308125 + 0.951346i \(0.400298\pi\)
\(444\) 0 0
\(445\) 3.65685 0.173352
\(446\) 0 0
\(447\) 15.6569 0.740544
\(448\) 0 0
\(449\) −15.6569 −0.738893 −0.369446 0.929252i \(-0.620453\pi\)
−0.369446 + 0.929252i \(0.620453\pi\)
\(450\) 0 0
\(451\) −46.6274 −2.19560
\(452\) 0 0
\(453\) −20.9706 −0.985283
\(454\) 0 0
\(455\) −2.82843 −0.132599
\(456\) 0 0
\(457\) −0.142136 −0.00664882 −0.00332441 0.999994i \(-0.501058\pi\)
−0.00332441 + 0.999994i \(0.501058\pi\)
\(458\) 0 0
\(459\) 0.828427 0.0386677
\(460\) 0 0
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) −6.14214 −0.285449 −0.142725 0.989762i \(-0.545586\pi\)
−0.142725 + 0.989762i \(0.545586\pi\)
\(464\) 0 0
\(465\) 1.65685 0.0768348
\(466\) 0 0
\(467\) −15.3137 −0.708634 −0.354317 0.935125i \(-0.615287\pi\)
−0.354317 + 0.935125i \(0.615287\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −21.3137 −0.982084
\(472\) 0 0
\(473\) −6.62742 −0.304729
\(474\) 0 0
\(475\) −2.82843 −0.129777
\(476\) 0 0
\(477\) −9.31371 −0.426445
\(478\) 0 0
\(479\) 10.3431 0.472590 0.236295 0.971681i \(-0.424067\pi\)
0.236295 + 0.971681i \(0.424067\pi\)
\(480\) 0 0
\(481\) 3.65685 0.166738
\(482\) 0 0
\(483\) −8.00000 −0.364013
\(484\) 0 0
\(485\) −1.51472 −0.0687798
\(486\) 0 0
\(487\) −5.17157 −0.234346 −0.117173 0.993111i \(-0.537383\pi\)
−0.117173 + 0.993111i \(0.537383\pi\)
\(488\) 0 0
\(489\) −16.9706 −0.767435
\(490\) 0 0
\(491\) −7.51472 −0.339135 −0.169567 0.985519i \(-0.554237\pi\)
−0.169567 + 0.985519i \(0.554237\pi\)
\(492\) 0 0
\(493\) −0.686292 −0.0309090
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) −22.6274 −1.01498
\(498\) 0 0
\(499\) −13.1716 −0.589641 −0.294820 0.955553i \(-0.595260\pi\)
−0.294820 + 0.955553i \(0.595260\pi\)
\(500\) 0 0
\(501\) 15.3137 0.684166
\(502\) 0 0
\(503\) 32.4853 1.44845 0.724224 0.689565i \(-0.242198\pi\)
0.724224 + 0.689565i \(0.242198\pi\)
\(504\) 0 0
\(505\) 6.48528 0.288591
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 1.31371 0.0582291 0.0291146 0.999576i \(-0.490731\pi\)
0.0291146 + 0.999576i \(0.490731\pi\)
\(510\) 0 0
\(511\) 8.97056 0.396834
\(512\) 0 0
\(513\) 2.82843 0.124878
\(514\) 0 0
\(515\) −2.34315 −0.103251
\(516\) 0 0
\(517\) 38.6274 1.69883
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −24.3431 −1.06649 −0.533246 0.845960i \(-0.679028\pi\)
−0.533246 + 0.845960i \(0.679028\pi\)
\(522\) 0 0
\(523\) 2.62742 0.114889 0.0574445 0.998349i \(-0.481705\pi\)
0.0574445 + 0.998349i \(0.481705\pi\)
\(524\) 0 0
\(525\) −2.82843 −0.123443
\(526\) 0 0
\(527\) 1.37258 0.0597907
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −11.6569 −0.504914
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) 8.48528 0.366167
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 11.4558 0.492525 0.246263 0.969203i \(-0.420797\pi\)
0.246263 + 0.969203i \(0.420797\pi\)
\(542\) 0 0
\(543\) −19.6569 −0.843556
\(544\) 0 0
\(545\) 2.48528 0.106458
\(546\) 0 0
\(547\) 0.686292 0.0293437 0.0146719 0.999892i \(-0.495330\pi\)
0.0146719 + 0.999892i \(0.495330\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −2.34315 −0.0998214
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) 3.65685 0.155225
\(556\) 0 0
\(557\) −27.9411 −1.18390 −0.591952 0.805973i \(-0.701642\pi\)
−0.591952 + 0.805973i \(0.701642\pi\)
\(558\) 0 0
\(559\) −1.65685 −0.0700775
\(560\) 0 0
\(561\) −3.31371 −0.139905
\(562\) 0 0
\(563\) −14.3431 −0.604492 −0.302246 0.953230i \(-0.597736\pi\)
−0.302246 + 0.953230i \(0.597736\pi\)
\(564\) 0 0
\(565\) −14.4853 −0.609400
\(566\) 0 0
\(567\) 2.82843 0.118783
\(568\) 0 0
\(569\) 14.2843 0.598828 0.299414 0.954123i \(-0.403209\pi\)
0.299414 + 0.954123i \(0.403209\pi\)
\(570\) 0 0
\(571\) −7.02944 −0.294173 −0.147086 0.989124i \(-0.546990\pi\)
−0.147086 + 0.989124i \(0.546990\pi\)
\(572\) 0 0
\(573\) 19.3137 0.806842
\(574\) 0 0
\(575\) 2.82843 0.117954
\(576\) 0 0
\(577\) 13.5147 0.562625 0.281313 0.959616i \(-0.409230\pi\)
0.281313 + 0.959616i \(0.409230\pi\)
\(578\) 0 0
\(579\) 21.7990 0.905935
\(580\) 0 0
\(581\) −43.3137 −1.79696
\(582\) 0 0
\(583\) 37.2548 1.54294
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) −6.34315 −0.261810 −0.130905 0.991395i \(-0.541788\pi\)
−0.130905 + 0.991395i \(0.541788\pi\)
\(588\) 0 0
\(589\) 4.68629 0.193095
\(590\) 0 0
\(591\) −9.31371 −0.383115
\(592\) 0 0
\(593\) 4.34315 0.178352 0.0891758 0.996016i \(-0.471577\pi\)
0.0891758 + 0.996016i \(0.471577\pi\)
\(594\) 0 0
\(595\) −2.34315 −0.0960596
\(596\) 0 0
\(597\) 16.9706 0.694559
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −2.68629 −0.109576 −0.0547881 0.998498i \(-0.517448\pi\)
−0.0547881 + 0.998498i \(0.517448\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) −5.65685 −0.229605 −0.114802 0.993388i \(-0.536623\pi\)
−0.114802 + 0.993388i \(0.536623\pi\)
\(608\) 0 0
\(609\) −2.34315 −0.0949491
\(610\) 0 0
\(611\) 9.65685 0.390675
\(612\) 0 0
\(613\) −13.0294 −0.526254 −0.263127 0.964761i \(-0.584754\pi\)
−0.263127 + 0.964761i \(0.584754\pi\)
\(614\) 0 0
\(615\) −11.6569 −0.470050
\(616\) 0 0
\(617\) 16.6274 0.669395 0.334697 0.942326i \(-0.391366\pi\)
0.334697 + 0.942326i \(0.391366\pi\)
\(618\) 0 0
\(619\) 34.8284 1.39987 0.699936 0.714205i \(-0.253212\pi\)
0.699936 + 0.714205i \(0.253212\pi\)
\(620\) 0 0
\(621\) −2.82843 −0.113501
\(622\) 0 0
\(623\) 10.3431 0.414389
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −11.3137 −0.451826
\(628\) 0 0
\(629\) 3.02944 0.120792
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) 16.0000 0.635943
\(634\) 0 0
\(635\) 2.34315 0.0929849
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −4.62742 −0.182772 −0.0913860 0.995816i \(-0.529130\pi\)
−0.0913860 + 0.995816i \(0.529130\pi\)
\(642\) 0 0
\(643\) −14.6274 −0.576849 −0.288425 0.957503i \(-0.593131\pi\)
−0.288425 + 0.957503i \(0.593131\pi\)
\(644\) 0 0
\(645\) −1.65685 −0.0652386
\(646\) 0 0
\(647\) −41.4558 −1.62980 −0.814899 0.579603i \(-0.803207\pi\)
−0.814899 + 0.579603i \(0.803207\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 4.68629 0.183670
\(652\) 0 0
\(653\) 38.2843 1.49818 0.749090 0.662469i \(-0.230491\pi\)
0.749090 + 0.662469i \(0.230491\pi\)
\(654\) 0 0
\(655\) −10.8284 −0.423102
\(656\) 0 0
\(657\) 3.17157 0.123735
\(658\) 0 0
\(659\) −4.20101 −0.163648 −0.0818241 0.996647i \(-0.526075\pi\)
−0.0818241 + 0.996647i \(0.526075\pi\)
\(660\) 0 0
\(661\) 4.82843 0.187804 0.0939020 0.995581i \(-0.470066\pi\)
0.0939020 + 0.995581i \(0.470066\pi\)
\(662\) 0 0
\(663\) −0.828427 −0.0321734
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 2.34315 0.0907270
\(668\) 0 0
\(669\) 2.82843 0.109353
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 6.28427 0.241524 0.120762 0.992681i \(-0.461466\pi\)
0.120762 + 0.992681i \(0.461466\pi\)
\(678\) 0 0
\(679\) −4.28427 −0.164415
\(680\) 0 0
\(681\) 6.34315 0.243070
\(682\) 0 0
\(683\) −29.9411 −1.14567 −0.572833 0.819672i \(-0.694156\pi\)
−0.572833 + 0.819672i \(0.694156\pi\)
\(684\) 0 0
\(685\) 13.3137 0.508691
\(686\) 0 0
\(687\) 8.82843 0.336826
\(688\) 0 0
\(689\) 9.31371 0.354824
\(690\) 0 0
\(691\) −34.8284 −1.32494 −0.662468 0.749090i \(-0.730491\pi\)
−0.662468 + 0.749090i \(0.730491\pi\)
\(692\) 0 0
\(693\) −11.3137 −0.429772
\(694\) 0 0
\(695\) −2.34315 −0.0888806
\(696\) 0 0
\(697\) −9.65685 −0.365779
\(698\) 0 0
\(699\) −26.4853 −1.00177
\(700\) 0 0
\(701\) −9.51472 −0.359366 −0.179683 0.983725i \(-0.557507\pi\)
−0.179683 + 0.983725i \(0.557507\pi\)
\(702\) 0 0
\(703\) 10.3431 0.390099
\(704\) 0 0
\(705\) 9.65685 0.363698
\(706\) 0 0
\(707\) 18.3431 0.689865
\(708\) 0 0
\(709\) −26.7696 −1.00535 −0.502676 0.864475i \(-0.667651\pi\)
−0.502676 + 0.864475i \(0.667651\pi\)
\(710\) 0 0
\(711\) −5.65685 −0.212149
\(712\) 0 0
\(713\) −4.68629 −0.175503
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 0 0
\(717\) −16.9706 −0.633777
\(718\) 0 0
\(719\) 37.6569 1.40436 0.702182 0.711998i \(-0.252209\pi\)
0.702182 + 0.711998i \(0.252209\pi\)
\(720\) 0 0
\(721\) −6.62742 −0.246818
\(722\) 0 0
\(723\) −21.3137 −0.792665
\(724\) 0 0
\(725\) 0.828427 0.0307670
\(726\) 0 0
\(727\) 14.6274 0.542501 0.271250 0.962509i \(-0.412563\pi\)
0.271250 + 0.962509i \(0.412563\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.37258 −0.0507668
\(732\) 0 0
\(733\) 1.02944 0.0380231 0.0190116 0.999819i \(-0.493948\pi\)
0.0190116 + 0.999819i \(0.493948\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −23.1127 −0.850214 −0.425107 0.905143i \(-0.639764\pi\)
−0.425107 + 0.905143i \(0.639764\pi\)
\(740\) 0 0
\(741\) −2.82843 −0.103905
\(742\) 0 0
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 0 0
\(745\) −15.6569 −0.573623
\(746\) 0 0
\(747\) −15.3137 −0.560299
\(748\) 0 0
\(749\) 11.3137 0.413394
\(750\) 0 0
\(751\) 51.3137 1.87246 0.936232 0.351383i \(-0.114288\pi\)
0.936232 + 0.351383i \(0.114288\pi\)
\(752\) 0 0
\(753\) 5.17157 0.188463
\(754\) 0 0
\(755\) 20.9706 0.763197
\(756\) 0 0
\(757\) 22.6863 0.824547 0.412274 0.911060i \(-0.364735\pi\)
0.412274 + 0.911060i \(0.364735\pi\)
\(758\) 0 0
\(759\) 11.3137 0.410662
\(760\) 0 0
\(761\) 14.9706 0.542682 0.271341 0.962483i \(-0.412533\pi\)
0.271341 + 0.962483i \(0.412533\pi\)
\(762\) 0 0
\(763\) 7.02944 0.254483
\(764\) 0 0
\(765\) −0.828427 −0.0299518
\(766\) 0 0
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) 35.9411 1.29607 0.648035 0.761610i \(-0.275591\pi\)
0.648035 + 0.761610i \(0.275591\pi\)
\(770\) 0 0
\(771\) 3.17157 0.114221
\(772\) 0 0
\(773\) −53.3137 −1.91756 −0.958780 0.284148i \(-0.908289\pi\)
−0.958780 + 0.284148i \(0.908289\pi\)
\(774\) 0 0
\(775\) −1.65685 −0.0595160
\(776\) 0 0
\(777\) 10.3431 0.371058
\(778\) 0 0
\(779\) −32.9706 −1.18129
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) −0.828427 −0.0296056
\(784\) 0 0
\(785\) 21.3137 0.760719
\(786\) 0 0
\(787\) 42.3431 1.50937 0.754685 0.656087i \(-0.227790\pi\)
0.754685 + 0.656087i \(0.227790\pi\)
\(788\) 0 0
\(789\) 14.1421 0.503473
\(790\) 0 0
\(791\) −40.9706 −1.45675
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) 9.31371 0.330323
\(796\) 0 0
\(797\) −12.6274 −0.447286 −0.223643 0.974671i \(-0.571795\pi\)
−0.223643 + 0.974671i \(0.571795\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 3.65685 0.129209
\(802\) 0 0
\(803\) −12.6863 −0.447690
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) 1.51472 0.0533206
\(808\) 0 0
\(809\) −25.3137 −0.889983 −0.444991 0.895535i \(-0.646793\pi\)
−0.444991 + 0.895535i \(0.646793\pi\)
\(810\) 0 0
\(811\) 15.1127 0.530679 0.265339 0.964155i \(-0.414516\pi\)
0.265339 + 0.964155i \(0.414516\pi\)
\(812\) 0 0
\(813\) 9.65685 0.338681
\(814\) 0 0
\(815\) 16.9706 0.594453
\(816\) 0 0
\(817\) −4.68629 −0.163953
\(818\) 0 0
\(819\) −2.82843 −0.0988332
\(820\) 0 0
\(821\) −14.2843 −0.498525 −0.249262 0.968436i \(-0.580188\pi\)
−0.249262 + 0.968436i \(0.580188\pi\)
\(822\) 0 0
\(823\) −5.65685 −0.197186 −0.0985928 0.995128i \(-0.531434\pi\)
−0.0985928 + 0.995128i \(0.531434\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 31.3137 1.08888 0.544442 0.838798i \(-0.316741\pi\)
0.544442 + 0.838798i \(0.316741\pi\)
\(828\) 0 0
\(829\) 41.3137 1.43488 0.717442 0.696618i \(-0.245313\pi\)
0.717442 + 0.696618i \(0.245313\pi\)
\(830\) 0 0
\(831\) 12.6274 0.438040
\(832\) 0 0
\(833\) −0.828427 −0.0287033
\(834\) 0 0
\(835\) −15.3137 −0.529953
\(836\) 0 0
\(837\) 1.65685 0.0572693
\(838\) 0 0
\(839\) −8.97056 −0.309698 −0.154849 0.987938i \(-0.549489\pi\)
−0.154849 + 0.987938i \(0.549489\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) 0 0
\(843\) 26.9706 0.928916
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 14.1421 0.485930
\(848\) 0 0
\(849\) 25.6569 0.880541
\(850\) 0 0
\(851\) −10.3431 −0.354558
\(852\) 0 0
\(853\) 34.9706 1.19737 0.598685 0.800985i \(-0.295690\pi\)
0.598685 + 0.800985i \(0.295690\pi\)
\(854\) 0 0
\(855\) −2.82843 −0.0967302
\(856\) 0 0
\(857\) −3.17157 −0.108339 −0.0541694 0.998532i \(-0.517251\pi\)
−0.0541694 + 0.998532i \(0.517251\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) −32.9706 −1.12363
\(862\) 0 0
\(863\) 20.0000 0.680808 0.340404 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) 0 0
\(867\) 16.3137 0.554043
\(868\) 0 0
\(869\) 22.6274 0.767583
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −1.51472 −0.0512655
\(874\) 0 0
\(875\) 2.82843 0.0956183
\(876\) 0 0
\(877\) 30.6863 1.03620 0.518101 0.855319i \(-0.326639\pi\)
0.518101 + 0.855319i \(0.326639\pi\)
\(878\) 0 0
\(879\) 10.0000 0.337292
\(880\) 0 0
\(881\) 10.9706 0.369608 0.184804 0.982775i \(-0.440835\pi\)
0.184804 + 0.982775i \(0.440835\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 4.00000 0.134459
\(886\) 0 0
\(887\) 44.7696 1.50321 0.751607 0.659611i \(-0.229279\pi\)
0.751607 + 0.659611i \(0.229279\pi\)
\(888\) 0 0
\(889\) 6.62742 0.222276
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 27.3137 0.914018
\(894\) 0 0
\(895\) −8.48528 −0.283632
\(896\) 0 0
\(897\) 2.82843 0.0944384
\(898\) 0 0
\(899\) −1.37258 −0.0457782
\(900\) 0 0
\(901\) 7.71573 0.257048
\(902\) 0 0
\(903\) −4.68629 −0.155950
\(904\) 0 0
\(905\) 19.6569 0.653416
\(906\) 0 0
\(907\) 49.6569 1.64883 0.824414 0.565987i \(-0.191505\pi\)
0.824414 + 0.565987i \(0.191505\pi\)
\(908\) 0 0
\(909\) 6.48528 0.215103
\(910\) 0 0
\(911\) −8.97056 −0.297208 −0.148604 0.988897i \(-0.547478\pi\)
−0.148604 + 0.988897i \(0.547478\pi\)
\(912\) 0 0
\(913\) 61.2548 2.02724
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) −30.6274 −1.01141
\(918\) 0 0
\(919\) 22.6274 0.746410 0.373205 0.927749i \(-0.378259\pi\)
0.373205 + 0.927749i \(0.378259\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) −3.65685 −0.120237
\(926\) 0 0
\(927\) −2.34315 −0.0769590
\(928\) 0 0
\(929\) −30.2843 −0.993595 −0.496797 0.867867i \(-0.665491\pi\)
−0.496797 + 0.867867i \(0.665491\pi\)
\(930\) 0 0
\(931\) −2.82843 −0.0926980
\(932\) 0 0
\(933\) 30.6274 1.00270
\(934\) 0 0
\(935\) 3.31371 0.108370
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) −10.9706 −0.358011
\(940\) 0 0
\(941\) −28.3431 −0.923960 −0.461980 0.886890i \(-0.652861\pi\)
−0.461980 + 0.886890i \(0.652861\pi\)
\(942\) 0 0
\(943\) 32.9706 1.07367
\(944\) 0 0
\(945\) −2.82843 −0.0920087
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) −3.17157 −0.102954
\(950\) 0 0
\(951\) −9.31371 −0.302018
\(952\) 0 0
\(953\) 47.7401 1.54645 0.773227 0.634129i \(-0.218641\pi\)
0.773227 + 0.634129i \(0.218641\pi\)
\(954\) 0 0
\(955\) −19.3137 −0.624977
\(956\) 0 0
\(957\) 3.31371 0.107117
\(958\) 0 0
\(959\) 37.6569 1.21600
\(960\) 0 0
\(961\) −28.2548 −0.911446
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) −21.7990 −0.701734
\(966\) 0 0
\(967\) −3.79899 −0.122167 −0.0610836 0.998133i \(-0.519456\pi\)
−0.0610836 + 0.998133i \(0.519456\pi\)
\(968\) 0 0
\(969\) −2.34315 −0.0752727
\(970\) 0 0
\(971\) 10.8284 0.347501 0.173750 0.984790i \(-0.444411\pi\)
0.173750 + 0.984790i \(0.444411\pi\)
\(972\) 0 0
\(973\) −6.62742 −0.212465
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) 41.5980 1.33084 0.665419 0.746470i \(-0.268253\pi\)
0.665419 + 0.746470i \(0.268253\pi\)
\(978\) 0 0
\(979\) −14.6274 −0.467494
\(980\) 0 0
\(981\) 2.48528 0.0793489
\(982\) 0 0
\(983\) −18.6274 −0.594122 −0.297061 0.954858i \(-0.596007\pi\)
−0.297061 + 0.954858i \(0.596007\pi\)
\(984\) 0 0
\(985\) 9.31371 0.296759
\(986\) 0 0
\(987\) 27.3137 0.869405
\(988\) 0 0
\(989\) 4.68629 0.149015
\(990\) 0 0
\(991\) −13.6569 −0.433824 −0.216912 0.976191i \(-0.569599\pi\)
−0.216912 + 0.976191i \(0.569599\pi\)
\(992\) 0 0
\(993\) 7.51472 0.238472
\(994\) 0 0
\(995\) −16.9706 −0.538003
\(996\) 0 0
\(997\) −2.68629 −0.0850757 −0.0425379 0.999095i \(-0.513544\pi\)
−0.0425379 + 0.999095i \(0.513544\pi\)
\(998\) 0 0
\(999\) 3.65685 0.115698
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6240.2.a.bn.1.2 2
4.3 odd 2 6240.2.a.bt.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.bn.1.2 2 1.1 even 1 trivial
6240.2.a.bt.1.1 yes 2 4.3 odd 2